Bernoulli numbers and the Pascal triangle

Taking advantage of the familiar expansion

we can write

From this it is clear that the function f(x) can be expanded in a power series about x=0 ; for the sake of convenience in subsequent computations, we represent this series as

where B₀=f(0)=1. In order to determine the other coefficients B_n (n=1,2, ... ) of the expansion, which are called Bernoulli numbers, we make use of the identity

Multiplying together the power series and equating to zero the coefficients of the positive powers of the variable x, we obtain an infinite system of linear equations :

or, multiplying by (n+1)! and noting that

If we agree to set

then last formula may be compactly written in the following symbolic form :

or, replacing (n+1) by n,

Putting n=2,3,4, .. in last formula, we obtain an infinite system of equations - connection with Bernoulli`s numbers and Pascal`s triangle :

Whence, we successively find

and where B₃=B₅=B₇= ... =B_2n+1=0.

Thus, the Bernoulli numbers may be determined step by step from the last symbolic formula; note that after the binomial expansion the powers of the B numbers must be replaced by Bernoulli numbers with the appropriate indices.